Grossness warning :)
Researchers have found that toilets produce an aerosol of the water in the toilet bowl, when they're flushed - review of research. These tiny droplets can dry out, then the germs remain airborne for awhile, and settle in various places in the bathroom, such as on one's toothbrush or on a roll of toilet paper.
So shutting the lid before flushing could be important. Toilets in hospitals etc. generally don't have lids, so this may allow infections to be transmitted.
One might get urinary tract infections this way, E. coli in the toilet water settling on the toilet paper or one's hands.